My wife brings her curling iron, so I don't see why you couldn't bring a straightener. Just keep in mind, that there is typically only one outlet in the cabin and its not in the bathroom (to avoid electric shock) So you can't have my than one going anyway. Just be aware that fire is much more dangerous at sea and remember to turn it off and make sure it is not able to touch anything flamable.
Carnival Cruise Lines:
Fantasy 90, Celebration 95, Victory 05, Conquest 05, Spirit 6/06, Freedom 6/08, Holiday 7/08, Victory 6/09, Valor 9/09, Dream 6/10, Splendor 8/10,Conquest 8/11,Magic 11/11,Triumph 4/12,Spirit 7/12,Freedom 11/12, Triumph 2/13 - The fire cruise, Magic 1/14, Freedom x 2 6/14
NCL Dream October 2006, RCCL Mariner OTS June 2007, Celebrity Silhouette 2013